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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Sufficiency classes of $\textrm {LCA}$ groups

Journal: Trans. Amer. Math. Soc. 158 (1971), 331-338
MSC: Primary 43A40
MathSciNet review: 0291728
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Abstract: By the sufficiency class $S(H)$ of a locally compact Abelian (LCA) group $H$ we shall mean the class of LCA groups $G$ having sufficiently many continuous homomorphisms into $H$ to separate the points of $G$. In this paper we determine the sufficiency classes of a number of LCA groups and indicate how these determinations may help to describe the structure of certain classes of LCA groups. In particular, we give a new proof of a theorem of Robertson which states that an LCA group is torsion-free if and only if its dual contains a dense divisible subgroup. We shall also derive some facts about the compact connected Abelian groups and a result about topological $p$-groups containing dense divisible subgroups. We conclude by giving a necessary condition for two LCA groups to have the same sufficiency class.

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Keywords: Locally compact Abelian, generalized character, sufficiency class, dual sufficiency class, divisible, reduced, torsion-free, compact and connected, topological <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img9.gif" ALT="$p$">-group, torsion subgroup
Article copyright: © Copyright 1971 American Mathematical Society